Curvature is mathematically defined as the inverse of the radius of curvature (K = 1/r).

On a two dimensional curve, the radius of curvature takes a single value. On a surface the radius of curvature depends on the direction and consequently, so does curvature.

On a surface, the curvature through a point can be calculated in any direction.

• For a dome-like shape, the curvature will be positive, regardless of the direction.
• For a valley, the curvature will be very different across the valley as compared to along the valley floor.
• For complicated saddle structures the curvature is positive in one direction and negative in another.

When analysing curvature, we are most often interested in the maximum curvature -- the largest magnitude curvature for a point, regardless of direction. This is not necessarily the most positive value -- a sharp valley will have negative curvature for the maximum attribute. For flat regions where dip is not changing, maximum curvature will be closer to zero.

At each point, the minimum curvature is the curvature measured in the direction perpendicular to the maximum. In the case of a valley, the maximum curvature will be measured across the valley, while the minimum curvature will be measured along the valley floor.

The maximum curvature highlights the biggest changes in dip. This is often a good fault indicator. When looking at horizon curvature at a fault boundary, there will be a strong positive curvature on the upthrown side, and a strong negative curvature on the downthrown side. Displaying the most positive (or most negative) curvature will show only one side of the fault, which can provide a clearer view for fault discrimination. The gap between the maximum positive and maximum negative curvature is an indication of the fault heave.

Maximum and minimum curvature together can describe the shape of a feature. Dome features have positive maximum and minimum curvature, and bowl features are described by negative values in all directions.

The shape index uses the following rules to classify features by curvature shape.

• +1 : Dome
  • Maximum curvature: positive
  • Minimum curvature: positive
• +0.5 : Ridge
  • Maximum curvature: positive
  • Minimum curvature: near zero
• 0: Flat plane or saddle:
  • Maximum curvature: near zero
  • Minimum curvature: near zero, OR
  • Max and min curvature have different signs
• -0.5: Valley or channel
  • Maximum curvature: negative
  • Minimum curvature: near zero
• -1 : Bowl:
  • Maximum curvature: negative
  • Minimum curvature: negative

Curvature shape does not take into account the amount of curvature. Use in combination with maximum curvature to identify the scale of the feature.

Curvedness or alternatively the "degree of flatness" can be calculated from the maximum and minimum curvature (the RMS average of the max and min curvature). It tends to be similar to the maximum curvature, with ridges and channels having relatively lower values than domes, bowls and saddles.

REFERENCES:

_{Al-Dosarry, Marfurt, 2006 - 3D volumetric multispectral estimates of reflector curvature and rotation. Geophysics, 2006, 71, 41-51.}

_{Roberts, A., 2001 - Curvature attributes and their application to 3D interpreted. Horizons: First Break, 2001, 19, 85–99.}